The importance of separable continuous trace \(C^*\)-algebras arises from the following facts: Firstly, their stable isomorphism classes are completely classifiable by topological data and, secondly, continuous-trace \(C^*\)-algebras form the building blocks of the more general type I \(C^*\)-algebras. This memoir presents an extensive study of strongly continuous actions of abelian locally compact groups on \(C^*\)-algebras with continuous trace. Under some natural assumptions on the underlying system \((A,G,\alpha )\), necessary and sufficient conditions are given for the crossed product \(A{\times }_{\alpha }G\) to have continuous trace, and some relations between the topological data of \(A\) and \(A{\times }_{\alpha }G\) are obtained. The results are applied to investigate the structure of group \(C^*\)-algebras of some two-step nilpotent groups and solvable Lie groups.

For readers' convenience, expositions of the Mackey-Green-Rieffel machine of induced representations and the theory of Morita equivalent \(C^*\)-dynamical systems are included. There is also an extensive elaboration of the representation theory of crossed products by actions of abelian groups on type I \(C^*\)-algebras, resulting in a new description of actions leading to type I crossed products.

Features:

• The most recent results on the theory of crossed products with continuous trace.
• Applications to the representation theory of locally compact groups and structure of group \(C^*\)-algebras.
• An exposition on the modern theory of induced representations.
• New results on type I crossed products.